PERCOLATION AND DISORDERED SYSTEMS Geoffrey GRIMMETT

156 Geoffrey Grimmett
3.1 Percolation Probability
3. PHASE TRANSITION
One of the principal objects of study is the percolation probability
(3.1) θ(p) = Pp(0 ↔ ∞),
or alternatively θ(p) = Pp(|C| = ∞) where C = C(0) is, as usual, the open
cluster at the origin. The event {0 ↔ ∞} is increasing, and therefore θ is
non-decreasing (using (2.4)), and it is natural to define the critical probability
pc = pc(L d ) by
pc = sup{p : θ(p) = 0}.
See Figure 1.1 for a sketch of the function θ.
3.2 Existence of Phase Transition
It is easy to show that pc(L) = 1, and therefore the case d = 1 is of limited
interest from this point of view.
Theorem 3.2. If d ≥ 2 then 0 < pc(L d ) < 1.
(3.3)
Actually we shall prove that
1
µ(d) ≤ pc(L d ) ≤ 1 − 1
µ(2)
where µ(d) is the connective constant of L d .
for d ≥ 2
Proof. Since L d may be embedded in L d+1 , it is ‘obvious’ that pc(L d ) is nonincreasing
in d (actually it is strictly decreasing). Therefore we need only to
show that
(3.4)
(3.5)
pc(L d ) > 0 for all d ≥ 2,
pc(L 2 ) < 1.
The proof of (3.4) is by a standard ‘path counting’ argument. Let N(n)
be the number of open paths of length n starting at the origin. The number
of such paths cannot exceed a theoretical upper bound of 2d(2d − 1) n−1 .
Therefore
� � � �
θ(p) ≤ Pp N(n) ≥ 1 ≤ Ep N(n)
≤ 2d(2d − 1) n−1 p n